The Generalized Liouville's Theorems via Euler-Lagrange Cohomology Groups on Symplectic Manifold

Abstract

Based on the Euler-Lagrange cohomology groups $H_{EL}^{(2k-1)}({\mathcal{M}}^{2n})(1⩽k⩽n)$ on symplectic manifold $({\mathcal{M}}^{2n},\omega )$, their properties and a kind of classification of vector fields on the manifold, we generalize Liouville's theorem in classical mechanics to two sequences, the symplectic(-like) and the Hamiltonian-(like) Liouville's theorems. This also generalizes Noether's theorem, since the sequence of symplectic(-like) Liouville's theorems link to the cohomology directly.

27 pages, no figure, revtex4